Sequential bezout space-time equalizers for MIMO systems

ABSTRACT

A receiver in a multiple-input-multiple-output, frequency-selective fading wireless communication systems sequentially recovers multiple data stream. A next input stream, having a highest signal-to-noise ratio is selected. The selected input stream is equalized, detected and decoded. The decoded data stream is then substracted from the data streams, and the selecting, equalizing, detecting and decoding, and subtracting is repeated until all of the data streams have been decoded.

FIELD OF INVENTION

[0001] The present invention relates generally to communicationssystems, and more particularly to interference cancellation and signalrecovery in wireless multiple-input-multiple-output communicationssystems.

BACKGROUND OF THE INVENTION

[0002] Multiple-input-multiple-output (MIMO) systems have the potentialto greatly increase the capacity of wireless communications systemswhere there are multiple antennas in both the transmitter and thereceiver.

[0003] A MIMO system has p transmitters and q receivers. If s_(j)(k) isa coded input sequence at transmitters j=1, . . . , p, h_(ij)(k) is achannel impulse response from transmitter j to receiver i=1, . . . , q,and d is a maximum length of the channel impulse response among all ofthe channels, then an output x_(i)(k) at receiver i can be expressed asa convolutional product $\begin{matrix}{{x_{i}(k)} = {{\sum\limits_{j = 1}^{p}{\sum\limits_{l = 0}^{d}{{h_{ij}(l)}{s_{j}( {k - l} )}}}} + {n_{i}(k)}}} & (1)\end{matrix}$

[0004] where n_(i)(k) denotes additive-white-Gaussian-noise (AWGN) atthe receiver i.

[0005] An equivalent expression of equation (1) in the frequency domainis

x(D)=H(D)s(D)+n(D)   (2)

[0006] where s(D)=[s₁(D) s₂(D) . . . s_(p)(D)]^(T), x(D)=[x₁(D) x₂(D) .. . x_(q)(D)]^(T), H(D)={H_(1j)(D)}_(1,j), and n(D)=[n₁(D) n₂(D) . . .n_(q)(D)]^(T) are the z-transform vectors (or matrix), and D=z⁻¹ denotesa unit delay of corresponding sequences or impulse responses.

[0007] The q×p polynomial matrix H(D) is referred to as the transferfunction of the MIMO system, and the polynomial vectorh_(j)(D)=[h_(1j)(D) h_(2j)(D) . . . h_(qj)(D)]^(T) (j=1, . . . p) is thechannel response from j^(th) transmitter antenna to all receiveantennas.

[0008] In such a wireless system, transmitted signal sequences aresubject to time-domain inter-symbol interference (ISI) and space-domaininter-channel-interference (ICI) from other signals. This makes itdifficult to correctly retrieve the transmitted sequences. In addition,for most practical channels, the frequency-response characteristics aretime-variant. This makes it more difficult to design an optimum filterand demodulator.

[0009] A Bezout equalizer offers an effective tool to reduce ISI and ICIin MIMO systems. The Bezout equalizer uses an array of linearfinite-impulse response (FIR) filters. To retrieve the input sequences{s_(j)(k)}_(j = 1)^(p)

[0010] from noise-corrupted observations {x_(i)(k)}_(i = 1)^(q),

[0011] the FIR filter can be applied at the receiver, see Ding et al.,“Blind Equalization and Identification,” Marcel Dekker, Inc., New York,2001. With appropriate parameters, a linear combination of the qfiltered receiver streams can reconstruct an individual input streamwhile reducing both ISI and ICI.

[0012] The following definitions are used for the Bezout inverse theory,set out below.

[0013] Definition 1—Perfect Recoverability

[0014] Given a MIMO channel with transfer function H(D), the j^(th)input is perfectly recoverable (PR) of order ρ if and only if thereexist a nonnegative integer k_(j) and a 1×q polynomial vector g(D) withdeg g(D)<ρ such that $\begin{matrix}{{{g(D)}{H(D)}} = {D^{k_{j}}e_{j}}} & (3)\end{matrix}$

[0015] where e_(j) is a unit (row) vector with all elements zero except1 at position j. The FIR filter array corresponding to g(D) in equation(3) is referred to as a (j, ρ, k) Bezout equalizer. The MIMO system issaid to be PR if and only if all the p inputs are PR of a finite order.

[0016] An expresion g(D) × (D) = s_(j)(D)D^(k_(j))

[0017] +noise term is obtained when g(D) in equation (3) is applied onthe receiver data yields, i.e. s_(j)(k) is reconstructed with noise anddelay k_(j). It is known that the condition of PR for a MIMO systemhinges upon the notion of coprimeness of the transfer function H(D), seeKailath et al., “Linear Systems,” Prentice-Hall, Englewood Cli., NJ,1980, and Kung et al., “An Associative Memory Approach to Blind SignalRecovery for SIMO/MIMO Systems,” IEEE Workshop on Neural Network forSignal Processing, September 2001.

[0018] Definition 2—Coprime Polynomial Matrices

[0019] A p×p polynomial matrix R(D) is said to be a right common divisorof the rows in H(D) if H(D)=H′(D)R(D), where H′(D) is itself apolynomial matrix. Furthermore, R(D) is called a greatest right commondivisor (grcd) if for any other right common divisor R′(D) there existsa polynomial matrix C(D) such that R(D)=C(D)R′(D). A polynomial matrixis delay-permissive right coprime if the determinant of its grcd has theform of a pure delay: det   R(D) = D^(k₀).

[0020] Theorem 1—PR Condition of MIMO System

[0021] A p-in-q-out MIMO system with transfer function H(D) is PR if andonly if H(D) is delay-permissive right coprime.

[0022] It is assumed that the channel transfer function is available atthe receiver end via some estimation procedure. For perfect recovery ingeneral, the coprime condition in Theorem 1 requires more receivers thantransmitters, i.e., q>p.

[0023]FIG. 1 shows a prior art parallel architecture of a MIMO system100. The system 100 includes transmitters 110, MIMO channel 120 subjectto noise 130, receivers 140, and Bezout equalizers 200. Here, s_(j)(k)111 are the inputs at the transmitters 110, x_(i)(k) 141 are the outputsat the receivers 140, and ŝ_(j)(k) 201 are the recovered inputs afterequalization 200. Under PR condition G(D)H(D) = Diag{D^(k_(j))}.

[0024]FIG. 2 shows the prior art Bezout equalizer 200 with FIRs 210. Thedesign of the Bezout equalizer can be decoupled into a task ofseparately designing individual equalizers for each input.

[0025] One prior art technique, which theoretically achieves channelcapacity in flat-fading MIMO systems, is called BLAST, see Foschini,“Layered Space-time Architecture for Wireless Communication in FadingEnvironments When Using Multiple Antennas,” Bell Labs Technical Journal,Vol. 1, pp.41-59, Autumn 1996. BLAST recognizes that flat-fading MIMOchannels, i.e., channels with multiple transmit and receive antennas,have enormous capacity. Capacity grows linearly with the number oftransmit antennas as long as the number of receiving antennas is greaterthan the number of transmitting antennas. The original BLAST used acyclic association of data streams, called layers, with transmitantennas, thereby producing an “averaged” channel which is the same forall layers. Difficulties in the realization of the original BLAST led toa modified architecture where each layer is associated with a certaintransmit antenna.

[0026] However, in order to achieve the full capacity of the MIMOchannel, long data blocks, powerful channel coding, and perfectdetection of each layer are required. In addition, in practical systems,the problem of error propagation limits the performance. Particularly,the overall diversity level is limited by the diversity level obtainedin the layer which is detected first. Most important, BLAST is onlyvalid for flat-fading channels, which limits its applicability tofrequency-selective channels in broadband communication.

[0027] Therefore, there is a need for a receiver in MIMO systems thatimprove upon the prior art.

SUMMARY OF THE INVENTION

[0028] The invention provides a system and method that combines Bezoutspace-time equalizers with sequential detection and decoding techniquesfor multiple-input-multiple-output (MIMO) communications systems. With asequential space-time equalizer, previously detected transmittingstreams are used to reduce interference in subsequent detected inputstream. The sequential equalization and detection/decoding according tothe invention successively reduces the number of unknown input streamsof the MIMO system. Excess dimensionality offered by the increasingasymmetry between the transmitted and received signal spaces providesthe necessary flexibility that improves the capacity of the system.

[0029] More particularly, the invention provides a method and system forequalizing signals transmitted over a multi-path channel and cancelingthe interference from the data streams sequentially. An input datastream with a highest post-processing signal-to-noise ratio (SNR) isrecovered first. The interference generated by this stream is thencancelled before detecting the stream with the next highest SNR. Thisprocedure is recursively executed until all the data streams have beenrecovered.

[0030] Furthermore, the invention provides a system and method thatprocesses the input sequences via a layered and pipeline architecture.

[0031] In the system and method according to the invention, twoadditional parameters are used: equalizer order and equalization delay.By selecting appropriate equalizer order and equalization delayparameters, the overall performance of the system can be optimized.

BRIEF DESCRIPTION OF THE DRAWINGS

[0032]FIG. 1 is a block diagram of a prior art parallel architecture ofa MIMO system;

[0033]FIG. 2 is a block diagram of a prior art Bezout equalizer;

[0034]FIG. 3 is a block diagram of a receiver according to theinvention;

[0035]FIG. 4 is a block diagram of sequential equalization,detection/decoding and cancellation according to the invention;

[0036]FIG. 5 is a block block diagram of a pipelined sequential Bezoutequalizer according to the invention; and

[0037]FIG. 6 is a block diagram of a layered pipeline sequential Bezoutequalizer according to the invention.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT OF THE INVENTION

[0038]FIG. 3 shows components of a receiver 300 in a MIMO system thatuses the invention. The components include a pre-processor 310, achannel estimator 320, and a sequential Bezout equalizer,detector/decoder, and interference canceller 400. The receiver 100 takesas input 301 signals received at multiple antennas, and produces asoutput 309 decoded data streams.

[0039] The operation of the receiver 300 is as follows. During thepre-processing 310, the input signals are filtered and time synchronizedto produce data streams for the Bezout Space-Time Equalizer. Channelimpulse response estimation is performed in block 320 to provide theH(D) 321 to the Bezout space-time equalizer 400. The functions of block400 are described in greater detail below.

[0040]FIG. 4 shows sequential equalization, detection/decoding andcancellation 400 according to the invention. This method yields a betterSNR or capacity in a MIMO system than obtainable with prior arttechniques.

[0041] First, select 405 a next input stream, of the j=1, . . . , p datastreams 401 stored in a memory 402, that has a highest post-processingsignal-to-noise ratio (SNR). Then, equalize 410 the selected data streamwith the Bezout FIR filter, the signal is then detected and decoded 420using an error-correction decoder. Next, cancel 430 the contribution ofthe detected stream 403 from the received data stored in the memory 402via a successive interference cancellation strategy as is commonly knownin signal processing. In essence, the cancellation 430 can be performedby subtracting the decoded signal from the received signal.

[0042] Repeat 440 the above steps of equalizing, detecting/decoding, andcancellation for a next input stream using the interference-reducedreceived signal 401, until all streams {s_(j)(k)}_(j) 409 have beendetected 450. Such a recursive method leads to a sequential Bezoutequalization strategy according to the invention.

[0043] Initially, we set H⁽¹⁾(D)=H(D), x⁽¹⁾(D)=x(D) and k₀=0. At step j,(j−1) input streams from transmitters have already been equalized,detected/decoded, and their interferences have been cancelled(subtracted) 430 from the receiver observation x(D) to obtain a new datavector, denoted as x^((j))(D). The operations in the j^(th) recursivestep are then: design an individual Bezout equalizer for stream j sothat $\begin{matrix}{{{g_{j}(D)}{H^{(j)}(D)}} = {D^{k_{j}}\lbrack {1\quad 0\quad \ldots \quad 0} \rbrack}} & (4)\end{matrix}$

[0044] and apply the equalizer on the recursively updated received datax^((j))(D) 401: $\begin{matrix}\begin{matrix}{{y_{i}(D)} = {{g_{j}(D)}{x^{(j)}(D)}}} \\{= {{D^{k_{1} + k_{2} + \ldots + k_{j}}{s_{j}(D)}} + {{noise}\quad {term}}}}\end{matrix} & (5)\end{matrix}$

[0045] Then, detect and decode the j^(th) selected stream witherror-correcting decoding 420 on y_(j)(D). Provided the coding schemehas sufficient error-correction ability, we obtain a correctreconstruction of the input sequence: ŝ_(j)(D)=s_(j)(D), with delay$\sum\limits_{i = 1}^{j}{k_{j}.}$

[0046] Cancel 430 the ICI generated by j^(th) input stream from thereceived observation vector based on the following recursive formulawhich basically is a subtraction:

x ^((j+1))(D)=D ^(k) ^(_(j)) x ^((j))(D)−D ^(k) ^(₁) ^(+ . . . +k)^(_(j)) h _(j)(D)ŝ _(j)(D)   (6)

[0047] Equation (6) represents a virtually truncated MIMO systemx^((j+1))(D)=D^(k) ^(₁) ^(+ . . . +) ^(_(j)) H^((j+1))(D)s^((j+1))(D)with

H ^((j+1))(D)=[h _(j+1)(D) . . . h _(p)(D)]

s ^((j+1))(D)=[s _(j+1)(D) . . . s _(p)(D)]^(T)   (7)

[0048] The reduced transfer function H^((j+1))(D) is the last (p−j)columns of H(D).

[0049] This procedure is recursively applied 440 until all the p inputsequences are decoded at the end 450. Each recursion results in asize-reduced MIMO system with one less input.

[0050]FIG. 5 shows a pipelined implementation 500 for realization of thesequential Bezout equalizer 400 according to the invention. There are players 501 in the pipeline 500 for recovering p data streams. Each layer501 includes the steps of equalization 410, detecting and decoding 420,and interference cancellation 430.

[0051] The layered and pipelined architecture 600 is shown in FIG. 6.The processing steps in each stage proceed from left to right. In thefirst stage, an individual input sequence is equalized 410 sequentiallyone block after the other with the temporal range of detected inputsymbols denoted by the labels on the blocks, e.g., N+1˜2N. Eachequalized block is then forwarded to the detector/decoder stage 420.Finally, the error-corrected sequence is used by the interferencecanceller (IC) stage 430 to cancel the interference contributed by thedetected sequence(s) from the receiver data. The interference-reduceddata are now ready to be processed in the next stage, as indicated bythe down arrows.

[0052] Each pipeline stage incurs an equalization delay of k_(j) forj=1, . . . , p together with a processing delay generated by the decoderand IC stages. The overall effect of these delays is depicted by theinter-layer block shifts with respect to processing time.

[0053] For two blocks with the same labeling, i.e., data blocks of twoinput streams within the same time interval, the block associated withthe lower stage is processed later time. In particular, the equalizationdelay generated by each individual Bezout equalizer is propagated to thenext stage through the decoder and IC stages, as shown by theinter-stage arrows 601. The interference-reduced received data at thebottom of j^(th) layer arrive at the (j+1)^(th) stage in the previousdata block, labeled by N+1˜2N, with exactly k_(j) symbols preceding thebeginning of the current block, labeled by 2N+1˜3N.

[0054] Although the lower (later) stages have a larger processing delay,they have a greater amount of estimated inputs obtained from the higher(earlier) stages. Consequently, assuming no error propagation, a largeramount of interference is cancelled from the received data by the laterstages. This, in turn, implies that the later stages are able to delivera higher SNR gain over the parallel scheme of the prior art.

[0055] Optimal Order of Signal Detection

[0056] To prevent error propagation in this sequential architecture, itis preferred to first recover the j*^(th) input stream whose individualBezout equalizer yields a highest post-processing 310 SNR. The detectionorder in the subsequent stages can then be determined in the samemanner.

[0057] The following process can be used for determining the order fordetecting the input streams.

[0058] Initially, set H⁽¹⁾(D)=H(D), and an input j*^(th) stream with ahighest SNR after pre-processing 310, see equation (12) below, isselected. Then, remove the j*^(th) column from H⁽¹⁾(D) to form atruncated system H⁽²⁾(D). This corresponds to the cancellation 430 ofinterference contributed by the j*^(th) input stream from the receiverdata. With the truncated transfer function H⁽²⁾(D), and itscorresponding individual Bezout equalizer design, the second stream isselected according to the same SNR criterion. This procedure isrecursively performed until all of the p data streams 409 have beendecoded.

[0059] A qρ×p(d+ρ) block Toeplitz resultant matrix is given below:$\begin{matrix}{{\Gamma^{\rho}\lbrack H\rbrack} = \begin{bmatrix}H_{0} & H_{1} & \ldots & H_{d} & 0 & \ldots & 0 \\0 & H_{0} & \ldots & H_{d - 1} & H_{d} & \ldots & 0 \\\vdots & \vdots & ⋰ & ⋰ & ⋰ & ⋰ & \vdots \\0 & 0 & \ldots & H_{0} & H_{1} & \ldots & H_{d}\end{bmatrix}} & (8)\end{matrix}$

[0060] where H_(i) denotes the i^(th) order coefficient matrix of thetransfer function H(D), i.e.,${H(D)} = {\sum\limits_{i = 0}^{d}{H_{i}{D^{i}\quad.}}}$

[0061] Due to the presence of the left null-space of H(D), there mayexist non-unique (j, ρ, k) Bezout equalizers satisfying equation (3). Atthe output of any equalizer g(D), the recovered signal preserves thepower of the j^(th) transmitting stream.

[0062] However, the i.i.d. AWGN in the receiver is filtered by g(D),leading to a post-processing noise power of${\frac{N_{0}}{2}{\overset{->}{g}}^{2}}\quad,$

[0063] where N₀ is the noise spectral density and {right arrow over(g)}=└g₀ g₁ . . . g_(ρ−1)┘ denotes the 1×qρ coefficient vector ofequalizer g(D). In order to maximize the post-processing SNR, one designcriterion minimizes the 2-norm of {right arrow over (g)}.

[0064] According to equation (3), an optimal (j, ρ, k) Bezout equalizer,if it exists, can be equivalently derived in a resultant matrix notationas: $\begin{matrix}{{\overset{->}{g}}^{*} = {\arg \quad {\min\limits_{\overset{->}{g}}\{ { {\overset{->}{g}}^{2} \middle| {\overset{->}{g}\quad {\Gamma^{\rho}\lbrack H\rbrack}}  = {\overset{->}{e}}_{r}} \}}}} & (9)\end{matrix}$

[0065] where {right arrow over (e)}_(r) is a row vector with allelements zero except 1 at r=j+pk_(j).

[0066] Given the transfer function H(D), equation (9) can be solved bytaking a singular value decomposition (SVD) on Γ^(ρ)[H]:

Γ^(ρ)[H]=UΣV   (10)

[0067] where Σ is a square diagonal matrix of positive singular values.Then, the solution to equation (9) is

{right arrow over (g)}*={right arrow over (e)} _(r) V ^(H)Σ⁻¹ U ^(H)  (11)

[0068] if and only if {right arrow over (e)}_(r)ε row span(Γ^(ρ[H]).)

[0069] Determination of j*. and k_(j)*

[0070] Given a predetermined equalizer order, the input streamassociated with the first stage can be selected via a joint optimizationof ∥{right arrow over (g)}*∥² in equation (11) over both the streamindex j and the equalization delay k_(j). As the pair (i, k_(j)) has aone-to-one correspondence with r=j+pk_(j), see equation (9), the samegoal can be achieved by minimizing ∥{right arrow over (g)}*∥² over r:$\begin{matrix}\begin{matrix}{r^{*} = {\arg \quad {\min\limits_{r}\{ ( {V^{H}\Sigma^{- 2}V} )_{rr} \middle| {{\overset{->}{e}}_{r} \in {{Row}\quad {Span}\{ V \}}} \}}}} \\{j^{*} = {\lbrack {( {r^{*} - 1} )\quad {mod}\quad p} \rbrack + 1}} \\{k_{j}^{*} = {d + \rho - 1 - \frac{r^{*} - j^{*}}{p}}}\end{matrix} & (12)\end{matrix}$

[0071] Thus equation (12) provides the optimal order for signaldetection. The same equation is used to determine the best recoverystream and equalization delay for every recursion or stage of thepipeline, upon replacement of H(D)in equation (9) and (10) by H^((l))(D)and p by p−l+1 in recursion l.

[0072] In the receiver according to the invention, each recursionreduces the dimension of the updated transfer function H^((i))(D) byone. This implies a reduced virtual MIMO channel with one less inputstream. Following the same idea as in equation (9), with the layereddetection procedure with ordering 1, 2, . . . p, the optimal individualBezout equalizer to recover input stream j is $\begin{matrix}{{\overset{->}{g}}^{*} = {\arg \quad {\min\limits_{\overset{->}{g}}\{ { {\overset{->}{g}}^{2} \middle| {\overset{->}{g}\quad {\Gamma^{\rho}\lbrack H^{(j)} \rbrack}}  = {\overset{->}{e}}_{r}} \}}}} & (13)\end{matrix}$

[0073] Because H^((j))(D) is a size-reduced version of H(D), all thenull-space solutions associated with the latter are also valid solutionsfor the former, but not vice versa. This means the post-processing SNRcorresponding to H^((j))(D) is equal or superior to H(D). In short, theSNR or capacity associated with the remaining source signals issignificantly enhanced.

[0074] Effect of the Invention

[0075] The receiver with the sequential Bezout equalizers according tothe invention has about double the SNR gain as that obtained by aparallel architecture of equal order. In addition, the receiver is lesssensitive to variations of equalization delay, which provides moreflexibility for recovery. For a fixed equalizer order, the sequentialarchitecture according to the invention has a much wider range withreasonable performance, while the parallel architecture degenerates morenoticeably around the optimal delay point.

[0076] This invention is described using specific terms and examples. Itis to be understood that various other adaptations and modifications maybe made within the spirit and scope of the invention. Therefore, it isthe object of the appended claims to cover all such variations andmodifications as come within the true spirit and scope of the invention.

We claim:
 1. A method for receiving a plurality of data streams in amultiple-input-multiple-output wireless communication systems,comprising: selecting a next input stream of the plurality of datastreams; equalizing the next input stream; detecting and decoding theequalized input stream; subtracting the decoded input stream from theplurality of data stream; and repeating the selecting, equalizing,detecting and decoding, and subtracting until all of the plurality ofdata streams have been decoded.
 2. The method of claim 1 wherein theselected input stream has a highest signal-to-noise ratio.
 3. The methodof claim 1 wherein the equalizing is performed by a Bezout equalizer. 4.The method of claim 1 further comprising: error-correcting whiledetecting and decoding.
 5. The method of claim 1 wherein the equalizing,detecting and decoding, and subtracting are pipelined with a pluralityof layers.
 6. The method of claim 5 wherein there is one layer for eachof the plurality of data streams.
 7. A receiver for receiving aplurality of data streams in a multiple-input-multiple-output wirelesscommunication systems, comprising: means for selecting a next inputstream of the plurality of data streams; an equalizer configured toequalize next input stream; a detector and decoder configured fordetecting and decoding the equalized input stream; means for subtractingthe decoded input stream from the plurality of data stream; and meansfor repeating the selecting, equalizing, detecting and decoding.
 8. Thereceiver of claim 7 wherein the means for selecting a next input stream,the equalizer, the detector and decoder, and the means for subtractingare pipelined.